2024/10/07 by Spacek, Peter, Wang, Charles
#05E10 #05E14 #14J33 #14M17 #14N35 #20G20 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2410.05070
We present a type-independent Landau-Ginzburg (LG) model (Xcan, Wcan) for any cominuscule homogeneous space X=G/P. We give a fully combinatorial construction for our superpotential Wcan as a sum of n+1 rational functions in the (generalized) Plücker coordinates on the "Langlands dual" minuscule homogeneous space \mathbbX=P^\vee\backslash G^\vee. Explicitly, we define the denominators Di_* of these rational functions using the combinatorics of order ideals of the corresponding minuscule poset, which can be interpreted as (generalized) Young diagrams, by a process that can be described by "moving boxes" and hence is easily implemented. To construct the corresponding numerators, we define derivations δi1 on ℂ[\mathbbX] that act by "adding an appropriate box if possible" and then we apply each δi1 to the corresponding Di_*. By studying certain Weyl orbits in the fundamental representations of \widetildeG^\vee and exploiting the existence of a certain dense algebraic torus in \mathbbX, we show that the polynomials Di_* coincide with the generalized minors ϕi_* appearing in the cluster structures for homogeneous spaces studied by Geiß-Leclerc-Schröer in arXiv:math/0609138. We then define the mirror variety Xcan=\mathbbX∖ Dac to be the complement of the anticanonical divisor Dac = ∑i_*\Di_*=0\ formed by the Di_*. Moreover, we show that the LG models (Xcan,Wcan) are isomorphic to the Lie-theoretic LG-models (XLie,WLie) constructed by Rietsch in arXiv:math/0511124 and our models naturally generalize the type-dependent Plücker coordinate LG-models previously studied by various authors.